ON SOLUTIONS OF THE NONHOMOGENEOUS CAUCHY PROBLEM FOR PARABOLIC TYPE DIFFERENTIAL EQUATIONS IN A BANACH SPACE

For a differential equation of the form $u'(t) + Au(t) = f(t), t \in (0,\infty)$, where $A$ is the infinitesimal generator of a bounded analytic $C_{0}$-semigroup of linear operators in a Banach space $\mathfrak{B}, \ f(t)$ is a $\mathfrak{B}$-valued polynomial, the behavior in the preassigned points of solutions of the Cauchy problem $u(0) = u_{0} \in \mathfrak{B}$ depending on $f(t)$ is investigated.